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Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains
On the variation of the fractional mean curvature under the effect of $C^{1, \alpha}$ perturbations
1. | Dipartimento di Matematica "Federigo Enriques", Università degli Studi di Milano, Via Saldini 50, I-20133 Milano, Italy |
References:
[1] |
L. Ambrosio, G. De Philippis and L. Martinazzi, Gamma-convergence of nonlocal perimeter functionals, Manuscripta Math., 134 (2011), 377-403.
doi: 10.1007/s00229-010-0399-4. |
[2] |
N. Abatangelo and E. Valdinoci, A notion of nonlocal curvature, Numer. Funct. Anal. Optim., 35 (2014), 793-815.
doi: 10.1080/01630563.2014.901837. |
[3] |
G. Albanese, A. Fiscella and E. Valdinoci, Gevrey regularity for integro-differential operators, J. Math. Anal. Appl., 428 (2015), 1225-1238.
doi: 10.1016/j.jmaa.2015.04.002. |
[4] |
G. Alberti and G. Bellettini, A nonlocal anisotropic model for phase transitions. Part I: The optimal profile problem, Math. Ann., 310 (1998), 527-560.
doi: 10.1007/s002080050159. |
[5] |
G. Alberti and G. Bellettini, A non-local anisotropic model for phase transitions: Asymptotic behaviour of rescaled energies, European J. Appl. Math., 9 (1998), 261-284.
doi: 10.1017/S0956792598003453. |
[6] |
B. Barrios, A. Figalli and E. Valdinoci, Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 13 (2014), 609-639. |
[7] |
X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.
doi: 10.1016/j.anihpc.2013.02.001. |
[8] |
X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions, Trans. Amer. Math. Soc., 367 (2015), 911-941.
doi: 10.1090/S0002-9947-2014-05906-0. |
[9] |
L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.
doi: 10.1002/cpa.20331. |
[10] |
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[11] |
L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638.
doi: 10.1002/cpa.20274. |
[12] |
L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal., 200 (2011), 59-88.
doi: 10.1007/s00205-010-0336-4. |
[13] |
L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations, 41 (2011), 203-240.
doi: 10.1007/s00526-010-0359-6. |
[14] |
L. Caffarelli and E. Valdinoci, Regularity properties of nonlocal minimal surfaces via limiting arguments, Adv. Math., 248 (2013), 843-871.
doi: 10.1016/j.aim.2013.08.007. |
[15] |
E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[16] |
S. Dipierro, A. Figalli, G. Palatucci and E. Valdinoci, Asymptotics of the s-perimeter as s ↘ 0, Discrete Contin. Dyn. Syst., 33 (2013), 2777-2790.
doi: 10.3934/dcds.2013.33.2777. |
[17] |
A. Figalli and E. Valdinoci, Regularity and Bernstein-type results for nonlocal minimal surfaces, J. Reine Angew. Math., published online, (2015).
doi: 10.1515/crelle-2015-0006. |
[18] |
N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, Berlin, 1972. |
[19] |
M. Ludwig, Anisotropic fractional perimeters, J. Differential Geom., 96 (2014), 77-93. |
[20] |
L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Ration. Mech. Anal., 98 (1987), 123-142.
doi: 10.1007/BF00251230. |
[21] |
L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza, Boll. Un. Mat. Ital. B (5), 14 (1977), 285-299. |
[22] |
G. Palatucci, O. Savin and E. Valdinoci, Local and global minimizers for a variational energy involving a fractional norm, Ann. Mat. Pura Appl., 192 (2013), 673-718.
doi: 10.1007/s10231-011-0243-9. |
[23] |
O. Savin and E. Valdinoci, $\Gamma$-convergence for nonlocal phase transitions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 479-500.
doi: 10.1016/j.anihpc.2012.01.006. |
[24] |
O. Savin and E. Valdinoci, Regularity of nonlocal minimal cones in dimension 2, Calc. Var. Partial Differential Equations, 48 (2013), 33-39.
doi: 10.1007/s00526-012-0539-7. |
[25] |
E. Valdinoci, A fractional framework for perimeters and phase transitions, Milan J. Math., 81 (2013), 1-23.
doi: 10.1007/s00032-013-0199-x. |
show all references
References:
[1] |
L. Ambrosio, G. De Philippis and L. Martinazzi, Gamma-convergence of nonlocal perimeter functionals, Manuscripta Math., 134 (2011), 377-403.
doi: 10.1007/s00229-010-0399-4. |
[2] |
N. Abatangelo and E. Valdinoci, A notion of nonlocal curvature, Numer. Funct. Anal. Optim., 35 (2014), 793-815.
doi: 10.1080/01630563.2014.901837. |
[3] |
G. Albanese, A. Fiscella and E. Valdinoci, Gevrey regularity for integro-differential operators, J. Math. Anal. Appl., 428 (2015), 1225-1238.
doi: 10.1016/j.jmaa.2015.04.002. |
[4] |
G. Alberti and G. Bellettini, A nonlocal anisotropic model for phase transitions. Part I: The optimal profile problem, Math. Ann., 310 (1998), 527-560.
doi: 10.1007/s002080050159. |
[5] |
G. Alberti and G. Bellettini, A non-local anisotropic model for phase transitions: Asymptotic behaviour of rescaled energies, European J. Appl. Math., 9 (1998), 261-284.
doi: 10.1017/S0956792598003453. |
[6] |
B. Barrios, A. Figalli and E. Valdinoci, Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 13 (2014), 609-639. |
[7] |
X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.
doi: 10.1016/j.anihpc.2013.02.001. |
[8] |
X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions, Trans. Amer. Math. Soc., 367 (2015), 911-941.
doi: 10.1090/S0002-9947-2014-05906-0. |
[9] |
L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.
doi: 10.1002/cpa.20331. |
[10] |
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[11] |
L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638.
doi: 10.1002/cpa.20274. |
[12] |
L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal., 200 (2011), 59-88.
doi: 10.1007/s00205-010-0336-4. |
[13] |
L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations, 41 (2011), 203-240.
doi: 10.1007/s00526-010-0359-6. |
[14] |
L. Caffarelli and E. Valdinoci, Regularity properties of nonlocal minimal surfaces via limiting arguments, Adv. Math., 248 (2013), 843-871.
doi: 10.1016/j.aim.2013.08.007. |
[15] |
E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[16] |
S. Dipierro, A. Figalli, G. Palatucci and E. Valdinoci, Asymptotics of the s-perimeter as s ↘ 0, Discrete Contin. Dyn. Syst., 33 (2013), 2777-2790.
doi: 10.3934/dcds.2013.33.2777. |
[17] |
A. Figalli and E. Valdinoci, Regularity and Bernstein-type results for nonlocal minimal surfaces, J. Reine Angew. Math., published online, (2015).
doi: 10.1515/crelle-2015-0006. |
[18] |
N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, Berlin, 1972. |
[19] |
M. Ludwig, Anisotropic fractional perimeters, J. Differential Geom., 96 (2014), 77-93. |
[20] |
L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Ration. Mech. Anal., 98 (1987), 123-142.
doi: 10.1007/BF00251230. |
[21] |
L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza, Boll. Un. Mat. Ital. B (5), 14 (1977), 285-299. |
[22] |
G. Palatucci, O. Savin and E. Valdinoci, Local and global minimizers for a variational energy involving a fractional norm, Ann. Mat. Pura Appl., 192 (2013), 673-718.
doi: 10.1007/s10231-011-0243-9. |
[23] |
O. Savin and E. Valdinoci, $\Gamma$-convergence for nonlocal phase transitions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 479-500.
doi: 10.1016/j.anihpc.2012.01.006. |
[24] |
O. Savin and E. Valdinoci, Regularity of nonlocal minimal cones in dimension 2, Calc. Var. Partial Differential Equations, 48 (2013), 33-39.
doi: 10.1007/s00526-012-0539-7. |
[25] |
E. Valdinoci, A fractional framework for perimeters and phase transitions, Milan J. Math., 81 (2013), 1-23.
doi: 10.1007/s00032-013-0199-x. |
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